Theorem kmlem9 8980

Description: Lemma for 5-quantifier AC of Kurt Maes, Th. 4, part of 3 => 4. (Contributed by NM, 25-Mar-2004.)

Hypothesis

Ref	Expression
kmlem9.1	⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}

Assertion

Ref	Expression
kmlem9	⊢ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)

Distinct variable groups:   𝑥,𝑧,𝑤,𝑢,𝑡   𝑧,𝐴,𝑤

Allowed substitution hints:   𝐴(𝑥,𝑢,𝑡)

Proof of Theorem kmlem9

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . 4 ⊢ 𝑧 ∈ V
2		eqeq1 2626	. . . . 5 ⊢ (𝑢 = 𝑧 → (𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
3	2	rexbidv 3052	. . . 4 ⊢ (𝑢 = 𝑧 → (∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ ∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
4		kmlem9.1	. . . 4 ⊢ 𝐴 = {𝑢 ∣ ∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))}
5	1, 3, 4	elab2 3354	. . 3 ⊢ (𝑧 ∈ 𝐴 ↔ ∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))
6		vex 3203	. . . . 5 ⊢ 𝑤 ∈ V
7		eqeq1 2626	. . . . . 6 ⊢ (𝑢 = 𝑤 → (𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ 𝑤 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
8	7	rexbidv 3052	. . . . 5 ⊢ (𝑢 = 𝑤 → (∃𝑡 ∈ 𝑥 𝑢 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ ∃𝑡 ∈ 𝑥 𝑤 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡}))))
9	6, 8, 4	elab2 3354	. . . 4 ⊢ (𝑤 ∈ 𝐴 ↔ ∃𝑡 ∈ 𝑥 𝑤 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})))
10		difeq1 3721	. . . . . . 7 ⊢ (𝑡 = ℎ → (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = (ℎ ∖ ∪ (𝑥 ∖ {𝑡})))
11		sneq 4187	. . . . . . . . . 10 ⊢ (𝑡 = ℎ → {𝑡} = {ℎ})
12	11	difeq2d 3728	. . . . . . . . 9 ⊢ (𝑡 = ℎ → (𝑥 ∖ {𝑡}) = (𝑥 ∖ {ℎ}))
13	12	unieqd 4446	. . . . . . . 8 ⊢ (𝑡 = ℎ → ∪ (𝑥 ∖ {𝑡}) = ∪ (𝑥 ∖ {ℎ}))
14	13	difeq2d 3728	. . . . . . 7 ⊢ (𝑡 = ℎ → (ℎ ∖ ∪ (𝑥 ∖ {𝑡})) = (ℎ ∖ ∪ (𝑥 ∖ {ℎ})))
15	10, 14	eqtrd 2656	. . . . . 6 ⊢ (𝑡 = ℎ → (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = (ℎ ∖ ∪ (𝑥 ∖ {ℎ})))
16	15	eqeq2d 2632	. . . . 5 ⊢ (𝑡 = ℎ → (𝑤 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))))
17	16	cbvrexv 3172	. . . 4 ⊢ (∃𝑡 ∈ 𝑥 𝑤 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ↔ ∃ℎ ∈ 𝑥 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ})))
18	9, 17	bitri 264	. . 3 ⊢ (𝑤 ∈ 𝐴 ↔ ∃ℎ ∈ 𝑥 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ})))
19		reeanv 3107	. . . 4 ⊢ (∃𝑡 ∈ 𝑥 ∃ℎ ∈ 𝑥 (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) ↔ (∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ ∃ℎ ∈ 𝑥 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))))
20		eqeq12 2635	. . . . . . . . . 10 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑧 = 𝑤 ↔ (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))))
21	15, 20	syl5ibr 236	. . . . . . . . 9 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑡 = ℎ → 𝑧 = 𝑤))
22	21	necon3d 2815	. . . . . . . 8 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑧 ≠ 𝑤 → 𝑡 ≠ ℎ))
23		kmlem5 8976	. . . . . . . . . 10 ⊢ ((ℎ ∈ 𝑥 ∧ 𝑡 ≠ ℎ) → ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) = ∅)
24		ineq12 3809	. . . . . . . . . . 11 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑧 ∩ 𝑤) = ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))))
25	24	eqeq1d 2624	. . . . . . . . . 10 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → ((𝑧 ∩ 𝑤) = ∅ ↔ ((𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∩ (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) = ∅))
26	23, 25	syl5ibr 236	. . . . . . . . 9 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → ((ℎ ∈ 𝑥 ∧ 𝑡 ≠ ℎ) → (𝑧 ∩ 𝑤) = ∅))
27	26	expd 452	. . . . . . . 8 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (ℎ ∈ 𝑥 → (𝑡 ≠ ℎ → (𝑧 ∩ 𝑤) = ∅)))
28	22, 27	syl5d 73	. . . . . . 7 ⊢ ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (ℎ ∈ 𝑥 → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))
29	28	com12 32	. . . . . 6 ⊢ (ℎ ∈ 𝑥 → ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))
30	29	adantl 482	. . . . 5 ⊢ ((𝑡 ∈ 𝑥 ∧ ℎ ∈ 𝑥) → ((𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)))
31	30	rexlimivv 3036	. . . 4 ⊢ (∃𝑡 ∈ 𝑥 ∃ℎ ∈ 𝑥 (𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))
32	19, 31	sylbir 225	. . 3 ⊢ ((∃𝑡 ∈ 𝑥 𝑧 = (𝑡 ∖ ∪ (𝑥 ∖ {𝑡})) ∧ ∃ℎ ∈ 𝑥 𝑤 = (ℎ ∖ ∪ (𝑥 ∖ {ℎ}))) → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))
33	5, 18, 32	syl2anb 496	. 2 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) → (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅))
34	33	rgen2a 2977	1 ⊢ ∀𝑧 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ∩ cin 3573  ∅c0 3915  {csn 4177  ∪ cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-uni 4437

This theorem is referenced by:  kmlem10  8981